4.12: Butcher tableaux
- Page ID
- 219169
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As mentioned above, there are many different variants of the Runge-Kutta method itself, as well as other types of predictor-corrector methods. A useful device which catalogs the different methods are the Butcher tableaux. These are tables summarizing the different coefficients which make up a particular method. A general, explicit predictor-corrector method looks like this:
\[\begin{aligned} k_1 =& h f(t_n, y_n) \\ k_2 =& h f(t_n + c_2 h, y_n + a_{21} k_1) \\ k_3 =& h f(t_n + c_3 h, y_n + a_{31} k_1 + a_{32} k_2) \\ & \vdots \\ y_{n+1} =& y_n + \left( b_1 k_1 + b_2 k_2 + b_3 k_3 + \cdots \right) \end{aligned} \label{eq:4.7}\]
Compare this to the definition of RK4 shown above. A Butcher tableau summarizes the coefficients as shown the [tab:4.1].
| \(c_1\) | \(a_{11}\) | \(a_{12}\) | \(a_{13}\) | \(\cdots\) |
| \(c_2\) | \(a_{21}\) | \(a_{22}\) | \(a_{23}\) | \(\cdots\) |
| \(c_3\) | \(a_{31}\) | \(a_{32}\) | \(a_{33}\) | \(\cdots\) |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\ddots\) |
| \(b_1\) | \(b_2\) | \(b_3\) | \(\cdots\) |
As examples, Butcher tableaux for Heun’s method, the midpoint method, and RK4 are shown in [tab:4.2] - [tab:4.4]. Compare the coefficients in the tables against the definitions of the solver methods to see how they are related.
| 0 | 0 | 0 |
| 1 | 1 | 0 |
| 1/2 | 1/2 |
\[\begin{aligned} \nonumber k_1 &= h f(t_n, y_n) \\ \nonumber k_2 &= h f(t_n+h, y_n + k_1) \\ \nonumber y_{n+1} &= y_n + k_1/2 + k_2/2 \end{aligned}\]
Table 4.2: Heun's method
| 0 | 0 | 0 |
| 1/2 | 1/2 | 0 |
| 0 | 1 |
\[\begin{aligned} \nonumber k_1 &= h f(t_n, y_n) \\ \nonumber k_2 &= h f(t_n+h/2, y_n + k_1/2) \\ \nonumber y_{n+1} &= y_n + k_2 \end{aligned}\]
Table 4.3: Midpoint method
| 0 | 0 | 0 | 0 | 0 |
| 1/2 | 1/2 | 0 | 0 | 0 |
| 1/2 | 0 | 1/2 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 1/6 | 1/3 | 1/3 | 1/6 |
\[\begin{aligned} \nonumber k_1 &= h f(t_n, y_n) \\ \nonumber k_2 &= h f(t_n+h/2, y_n + k_1/2) \\ \nonumber k_3 &= h f(t_n+h/2, y_n + k_2/2) \\ \nonumber k_4 &= h f(t_n+h, y_n + k_3) \\ \nonumber y_{n+1} &= y_n + (k_1 + 2 k_2 + 2 k_3 + k_4)/6 \end{aligned}\]
Table 4.4: Fourth-order Runge-Kutta